The present invention is related to capacitors.
As the speed of systems and circuits continues to increase, particularly of integrated circuits, associated packaging elements, and wiring of printed circuit panels to which they are attached, prior assumptions made concerning the behavior of certain circuit elements do not always hold true. At higher frequencies, the alternating current (AC) behavior of such circuit elements begins to predominate over that which is expected at lower frequencies.
At speeds that are currently being achieved not only within an integrated circuit (“IC” or “chip”), but off the chip as well, specific consideration must be given to the design and interconnection of circuit elements to maximize their performance at the frequencies they are expected to operate.
Particularly with respect to capacitors, prior assumptions about their behavior can lead to unsatisfactory results at high frequency. As shown in FIG. 1, a capacitor has an associated equivalent series resistance R12 and an equivalent parallel resistance R14, as well as an ideal capacitance C12. At DC or low frequencies, the impedance of the capacitor due to these resistances R12 and R14 appears small in relation to the impedance due to the capacitance C12, as such impedance is determined by the relation Z(C)=1/jωC. However, when operating at higher frequencies, the impedance due to the capacitance Z(C) decreases, such that the resistances of R12 and R14 become proportionately larger.
With increasing frequency, not only does the impedance Z(C) of the capacitor C12 decrease, but the equivalent series resistance R12 due to the resistance of the capacitor's conductive plates actually increases. All conductive materials, other than ideal superconductive materials, are known to have at least some resistivity. In addition to resistance which is apparent at a steady current DC, such resistivity arises from electromagnetic effects which can cause radiative as well as thermal losses.
Thus, in AC operation, certain dielectric materials used in a capacitor are known to cause thermal losses, in which the capacitor dissipates the energy of an electric field by heating. In general, dielectric materials having a high dielectric constant K, such as tantalum pentoxide and aluminum oxide, tend to cause higher thermal losses than dielectric materials which have lower dielectric constants. Moreover, the losses are more pronounced at higher frequencies. Referring to FIG. 1, such losses are manifested as a component of the resistance R12 in series with the capacitor and the resistance R14 in parallel with the capacitor.
Another factor contributing to equivalent series resistance is the skin effect. Skin effect is the tendency of AC currents to travel within the outermost layer of a conductor. As manifested by the skin effect, electromagnetic fields, and the resulting current density, decay exponentially in relation to the depth from the surface of a conductor. The higher the frequency, the more pronounced the skin effect becomes, leading to a shallower “skin depth” within which most of the AC current is conducted, and below which, comparatively little AC current is conducted.
The skin depth δ is defined by the relation:
  δ  ≡            c                        2          ⁢                                          ⁢          πσμω                      .  In this equation, ω is the frequency, c is the speed of light in free space, σ the conductivity of the conductor, and μ is the permeability of the conductor.
Thus, the skin depth δ is inversely proportional to the square root of the frequency ω, the conductivity σ and the permeability μ of the conductor. Hence, a better conductor has a smaller skin depth, and the skin depth decreases with frequency, i.e. varies inversely with frequency.
Resistance is related to skin depth in the following way. The resistance of an object, be it a conductor or otherwise, is defined as the inverse of its conductivity, i.e., the resistivity, multiplied by the length, and divided by the cross-sectional area of the object through which current passes; i.e. R=ρ×L/(Wt), wherein ρ is the resistivity, L is the length of the conductor, and W and t are the width and thickness of the conductor defining the cross-sectional area, respectively. For higher frequencies, the skin depth is a measure of the effective thickness of the conductor at a particular frequency.
Shallower skin depth reduces the cross-sectional area through which currents pass within the conductor, as the current mostly passes within the skin depth. Since resistance is inversely related to the cross-sectional dimensions of the conductor, a conductor's resistance increases as the skin depth decreases at higher frequencies.
Sheet resistance Rs, measured in terms of ohms per square, is related to the resistance of an object by the particular geometry of the object through which the current passes. In any path capable of conducting a current, the resistance is equal to the sheet resistance times the ratio of the length of the path to the width of the path,i.e., R=Rs×(L/W)According to the above relation, the longer the distance the current must travel in a particular path, the higher the resistance will be. Also, the narrower the width of the path, the higher the resistance will be.
As described above, the trend toward higher operational frequencies, and the resulting decreases in skin depth tends to make the plates of a capacitor more resistive. In addition, the use of certain high-K dielectric materials and conductors other than copper or aluminum is also leading to increases in the resistance of capacitor plates.
As certain high-K dielectric materials are not compatible with the most conductive metals such as copper and aluminum, the use of lower conductivity metals in capacitors having high-K dielectrics further increases the resistance. For example, while copper has a conductivity of 6.0×107/Ω.m, platinum, which can be combined in capacitors having some types of high-K ferroelectric dielectrics, has conductivity of only 9.6×106/Ω.m, only about one sixth that of copper.
Heretofore, little consideration has been given to the geometry of the interconnection between the plates of a capacitor and other elements in a way which reduces the equivalent series resistance of the capacitor. Because of the higher frequencies at which circuits now operate, not only in chips, but in packaging and circuit panel elements as well, and the more lossy (resistive) types of materials used in capacitors, the resistance of a capacitor has become significant relative to the capacitance, such that a new way is needed to lower the equivalent series resistance.
The failure of the prior art to address this concern is best explained with reference to the example shown in FIGS. 2A and 2B. Conventionally, as shown in the plan view of FIG. 2A and the cross-sectional view through lines 2B—2B thereof in FIG. 2B, a capacitor 18 having two plates 20 separated by a capacitor dielectric 22 is coupled to other circuit elements through a conductor, which is formed as a central post 24 in the center of plate 20, as shown in FIGS. 2A–2B. The other plate 20 of the capacitor may either be connected to other circuit elements through another central post 26, or the plate 20 may coincide with a ground plane, or, alternatively, the plane of a power supply.
As noted above, the resistance of a current path is directly related to the ratio of the length to width of the path in which the current is conducted. Thus, as shown in FIGS. 2A–2B, in a capacitor having square plates 20 measuring two units on a side, the distance that currents cross from the central post 24 to a north or south edge 28 of the plate is one unit, and from the central post 24 to the east or west edge 30 of the plate is also one unit. Such distance, being in the direction from an origin (the center post 24) to the edge (28 or 30), represents the length of the path, for purposes of determining the resistance. However, the distance from the center of the post 24 to the corners 32 of the plate 20 is not one unit, but rather √2. Thus, in the square plate capacitor 18 of FIGS. 2A and 2B, the current must travel a minimum length of one unit from the central post 24 to the edges 28 and 30, while a length of up to √2 may have to be crossed to reach the corners of the capacitor plate 20.
One possible way of reducing the path length would be to form the capacitor plates as circular disks 34, as shown in FIG. 3, such that the length of the distance the current crosses from a central post 36 to the edge 38 would be uniformly one unit. However, such shape might not be easily provided in an integrated circuit, packaging element or circuit panel, as processing is generally optimized for the formation of generally rectangular circuit elements rather than circular elements. In addition, the space within a rectangular region 40 surrounding the capacitor 34 might end up being wasted. Thus, disk geometry does not solve the above problem in a satisfactory way.
The second factor, width of the path of the current must also be considered, since resistance is inversely related to the width. In this context, the width is the dimension through which the current passes at one time. With reference to FIG. 2A, although the plate capacitor 20 has square shape, the direction of current flow is radially outward from the central post 24. Thus, the width of the current path at any point in time is the circumference of a circle having a radius 42 equal to the distance from the center of the central post 24. As the distance from the center increases, the width increases by a factor of 2π. Consequently, when a central post 24 is used that is relatively small in relation to the size of the capacitor plate 20 (e.g. 1/10), the resistance will be many times higher near the central post 24 than near the edges 28, 30 of the plate 20. Accordingly, the width of the central post 24 that interconnects the capacitor 18 to other circuitry becomes a limiting factor in reducing the series resistance of the capacitor 18.
For these and other reasons, therefore, there has been an increasing need for a capacitor design which will mitigate the effects of resistivity, particularly in high-frequency operation.